Markfelder S. Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations

Cham: Springer, 2021. — 244 p.This book applies the convex integration method to multi-dimensional compressible Euler equations in the barotropic case as well as the full system with temperature. The convex integration technique, originally developed in the context of differential inclusions, was applied in the groundbreaking work of De Lellis and Székelyhidi to the incompressible Euler equations, leading to infinitely many solutions. This theory was later refined to prove non-uniqueness of solutions of the compressible Euler system, too. These non-uniqueness results all use an ansatz which reduces the equations to a kind of incompressible system to which a slight modification of the incompressible theory can be applied. This book presents, for the first time, a generalization of the De Lellis–Székelyhidi approach to the setting of compressible Euler equations.
The structure of this book is as follows: after providing an accessible introduction to the subject, including the essentials of hyperbolic conservation laws, the idea of convex integration in the compressible framework is developed. The main result proves that under a certain assumption there exist infinitely many solutions to an abstract initial boundary value problem for the Euler system. Next some applications of this theorem are discussed, in particular concerning the Riemann problem. Finally there is a survey of some related results.
This self-contained book is suitable for both beginners in the field of hyperbolic conservation laws as well as for advanced readers who already know about convex integration in the incompressible framework.The Problem Studied in This BookThe Euler Equations
Weak Solutions and Admissibility
Overview on Well-Posedness Results
Structure of This Book
Hyperbolic Conservation Laws
Formulation of a Conservation Law
Initial Boundary Value Problem
Hyperbolicity
Companion Laws and Entropies
Admissible Weak Solutions
The Euler Equations as a Hyperbolic Systemof Conservation Laws
Barotropic Euler System
Hyperbolicity
Entropies
Admissible Weak Solutions
Full Euler System
Hyperbolicity
Entropies
Admissible Weak SolutionsConvex IntegrationPreparation for Applying Convex Integrationto Compressible Euler
Outline and Preliminaries
Adjusting the Problem
Tartar's Framework
Plane Waves and the Wave Cone
Sketch of the Convex Integration Technique
-Convex Hulls
Definitions and Basic Facts
The HN-Condition and a Way to Define U
The -Convex Hull of Slices
The -Convex Hull if the Wave Cone is Complete
The Relaxed Set U Revisited
Definition of U
Computation of U
Operators
Statement of the Operators
Lemmas for the Proof of Proposition
Proof of Proposition
Implementation of Convex Integration
The Convex-Integration-Theorem
Statement of the Theorem
Functional Setup
The Functionals I and the Perturbation Property
Proof of the Convex-Integration-Theorem
Proof of the Perturbation Property
Lemmas for the Proof
Proof of Lemma
Proof of Lemma Using Lemmas , and
Proof of the Perturbation Property Using Lemma
Convex Integration with Fixed Density
A Modified Version of the Convex-Integration-Theorem
Proof the Modified Perturbation PropertyApplication to Particular Initial (Boundary) Value ProblemsInfinitely Many Solutions of the Initial Boundary Value Problem for Barotropic Euler
A Simple Result on Weak Solutions
Possible Improvements to Obtain Admissible Weak Solutions
Further Possible Improvements
Riemann Initial Data in Two Space Dimensionsfor Isentropic Euler
One-Dimensional Self-Similar Solution
Summary of the Results on Non-/Uniqueness
Non-Uniqueness Proof if the Self-Similar Solution Consists of One Shock and One Rarefaction
Condition for Non-Uniqueness
The Corresponding System of Algebraic Equations and Inequalities
Simplification of the Algebraic System
Solution of the Algebraic System if the Rarefaction is ``Small''
Proof of Theorem via an Auxiliary State
Sketches of the Non-Uniqueness Proofs for the Other Cases
Two Shocks
One Shock
A Contact Discontinuity and at Least One Shock
Other Results in the Context of the Riemann Problem
Riemann Initial Data in Two Space Dimensions for Full Euler
One-Dimensional Self-Similar Solution
Summary of the Results on Non-/Uniqueness
Non-Uniqueness Proof if the Self-Similar Solution Contains Two Shocks
Condition for Non-Uniqueness
The Corresponding System of Algebraic Equations and Inequalities
Solution of the Algebraic System
Sketches of the Non-Uniqueness Proofs for the Other Cases
One Shock and One Rarefaction
One Shock
Other Results in the Context of the Riemann Problem
Notation and Lemmas
Sets
Vectors and Matrices
General Euclidean Spaces
The Physical Space and the Space-Time
Phase Space
Sequences
Functions
Basic Notions
Differential Operators
Functions of Time and Space
Functions of the State Vector
Function Spaces
Integrability Conditions
Boundary Integrals and the Divergence Theorem
Mollifiers
Periodic Functions
Convexity
Convex Sets and Convex Hulls
Convex Functions
Semi-Continuity
Weak- Convergence in L∞
Baire Category TheoremError Report